Ordinal Numbers

Ordinal numbers (a.k.a. ordinals) generalize the notion of "labels" of objects in an ordered set. The collection of ordinals can be thought of as a "universal label collection," in the sense that any well-behaved order on a set is describable using a sequence of ordinals.

This image depicts the ordinals, showing how \(\omega\) is larger than every integer, and showing the names of ordinals larger than \(\omega\).

Contents

Motivation

For example, the natural numbers \(\mathbb{N}\) have a standard ordering, which is expressed by the sequence of labels \(\{1, 2, 3, \cdots\}\). The integers double as both elements of \(\mathbb{N}\) and as labels for the order on \(\mathbb{N}\); following the latter sense, one thinks of \(\{1, 2, 3, \cdots\}\) as the finite ordinals. This is because any order on any finite set \(S\), say of size \(n\), can be described by a bijection \(f: \{1, 2, \cdots, n\} \to S\); the bijection induces the order \(\le_{S}\) on \(S\) given by \(a\le_{S} b\) iff \(f^{-1} (a) \le f^{-1} (b)\).

Similarly, any order on \(\mathbb{N}\) can be described using only finite ordinals. However, if an infinite set \(S\) has larger cardinality than \(\mathbb{N}\), the labels \(\{1, 2, 3, \cdots\}\) would not suffice to describe the entire order on \(S\). Accordingly, one introduces the infinite ordinal \(\omega\), which is constructed to be larger than any finite ordinal. Then, an element of \(S\) which is larger in the ordering \(\le_{S}\) than any of the elements labeled by \(\{1, 2, 3, \cdots\}\) could be labeled with \(\omega\).

Preliminaries: Ordering Sets

Let \(S\) be a set. A total order on \(S\) is a binary relation, denoted by \(\le \), such that for all \(a, b, c \in S\), the following axioms hold:

Either \(a\le b\) or \(b \le a\).

If \(a\le b \) and \(b\le a\), then \(a=b\).

If \(a\le b\) and \(b\le c\), then \(a\le c\).

For example, the standard ordering on \(\mathbb{R}\) is a total order. As a consequence, the dictionary order on \(\mathbb{R}^n\) is also a total order; this is defined by setting \[(x_1, x_2, \cdots, x_n) \le (y_1, y_2, \cdots, y_n)\] iff \(x_i \le y_i\) in the standard ordering on \(\mathbb{R}\) for all \(1\le i \le n\).

A total order \(\le \) on \(S\) is called a well order if it permits no infinite decreasing sequence of elements in \(S\). Equivalently, \(\le \) is a well order iff every non-empty subset of \(S\) has a least element in the order \(\le\). Note that \(\mathbb{R}\) with the standard ordering is not well-ordered, since for every element \(x\in \mathbb{R}\), the element \(y:= x-1 \in \mathbb{R}\) satisfies \(y \le x\). However, the natural numbers \(\mathbb{N}\) with their natural ordering are well-ordered, by the well-ordering principle.

Two sets \(A\) and \(B\), with total orders \(\le_{A}\) and \(\le_{B}\) respectively, are called order-isomorphic if there exists a bijection \(f: A \to B\) such that \(a \le_{A} b\) implies \(f(a) \le_{B} f(b)\) for all \(a,b \in A\).

Let \(A\) be the number of totally-ordered sets below, and let \(B\) be the number of well-ordered sets below. What is \(A+B\)?

\(S = \mathbb{R}\) with the standard order.

\(S = 2^{\mathbb{R}}\), the power set of \(\mathbb{R}\), ordered by set inclusion, i.e. \(A \le B\) iff \(A \subseteq B\).

Constructing Ordinal Numbers

Suppose that a sequence of ordinal labels has been constructed. Then, every ordinal \(x\) describes a well-ordered set, namely the set of all ordinals less than \(x\). This set, which we also denote by \(x\), can be thought of as the canonical example of a particular well-ordered set, in the sense that it represents the equivalence class of all well-ordered sets that are order-isomorphic to it.

To illustrate this, consider the finite ordinal \(5\), which one identifies with the ordered set of ordinals less than it, namely \(5 = \{1, 2, 3, 4\}\). Then, any well-ordered four element set is order-isomorphic to \(5\); for example, the four element set \(\{a,b,c,d\}\) with well order \(b\le a \le d\le c\) permits the order-isomorphism \(f: 5 \to \{a,b,c,d\}\) given by \(f(1) = b\), \(f(2) = a\), \(f(3) = d\), and \(f(4) = c\).

Based on this intuition, one formally defines ordinals as equivalence classes of well-ordered sets. If \((A, \le _{A})\) and \((B, \le_{B})\) are well-ordered sets, then one writes \(A \sim B\) if \(A\) and \(B\) are order-isomorphic; this relation \(\sim\) is an equivalence relation, and the equivalence classes obtained are called ordinals. Denote by \([A]\) the equivalence class of \(A\), i.e. the ordinal containing the well-ordered set \(A\).

In this formalism, the finite ordinals are precisely \(1:= [\emptyset], 2:= [\{1\}], 3:= [\{1, 2\}], 4:=[\{1,2,3\}], \cdots\). The smallest infinite ordinal is \(\omega := [\mathbb{N}]\). For an example of an infinite ordinal larger than \(\omega\), consider the set \(S:= \mathbb{N} \cup \{\star\}\) with well order \[1 \le 2 \le 3 \le \cdots \le \star.\] Since \(S\) does not have the same cardinality as \(\mathbb{N}\), the ordinal \([S]\) does not equal \(\omega\). Usually, one denotes \(\omega + 1 := [S]\).